A recent paper from Tel Aviv regarding Egyptian standard sizes for round pottery jugs may have inadvertently helped to solve an Egyptian mathematical mystery:
A cursory examination of the new information suggests it is significant and may provide a long-sought-after explanation for the seeming existence of two different measurement traditions used for circular calculations in Egypt: one algorithmic, used in papyri and based on a diameter of 9 units (i.e. 28/9), and one geometric and used in monumental architecture and based on a circumference of 3+1/7th times the diameter (i.e. 22/7) (Legon 1990; Legon 1991; Lightbody 2008; Cooper 2011: 478; Lightbody 2012).
Like Cooper, Legon and Petrie, I have argued (Lightbody 2008: 54) that two different methods for handling circular calculations were known to the Ancient Egyptians, and that textual evidence for one does not negate the archaeological evidence for the use of the other. Both systems were vernacular and had their different strengths and weaknesses, and were utilised according to the applications and requirements.
The new information seems to explain how one volume measurement system developed out the use of a 1 cubit circumference near-spherical jars as a common standard volume for liquids. The volume of this common size works out at half a ‘hekat’, the Egyptian’s measure of volume (half a hekat is approximately 2.4 liters). This neat relationship, along with evidence from the Rhind Mathematical Papyrus an Moscow Mathematical Papyrus, may explain where the Egyptian method of circular area and volume calculations based on the number 9 stemmed from. One cubit is 28 digits, and this standard circumference gives an approx diameter of 9 digits. Evidence from elsewhere suggests that they were aware of this approximate relationship and used it. Moscow Mathematic Papyrus example 10 gives an example of a hemispherical basket area calc based on a 4.5 units diameter, being half of 9. Likewise Rhind Mathematical Papyrus examples 50 and 48 show circular area calculations based on subtracting 1/9th of diameters and squaring the remainder to get the required result. These examples suggest they were used to working on diameters of 9 and therefore 28 digits circumference which is 1 cubit (Gillings 1982: 140).
Archaeological and architectural evidence (columns, pyramids, tombs (Petrie 1892; Petrie 1925; Petrie 1940: 30; Edwards 1979: 269; Verner 2003: 70; Lightbody 2008: 18; Cooper 2011)), however, suggests that the Egyptians also used 22/7 (3+1/7th) as an architectural approximation for circumferences, yet they did not use this in surviving area and volume calcs on papyri. I have argued that Egyptians were not aware that a circumference/diameter ratio could be extended to area and volume calcs as we do now, and so could not use the 22/7 (3+1/7th) approximation for these. This new information also suggests that the alternative 28/9 approximation for circumference/diameter arose from a common standard size that was already in use during the Old Kingdom (according to the new article), and that a handy ad-hoc area and volume system developed out of those numbers and from the standard cubit.
In conclusion, the more accurate architectural circumference calculation system based on 22/7 was not easily applicable to area and volume calculations, and would therefore not necessarily have superseded an existing one based on 28/9 that was fairly accurate, functional and easy to use. This explains the apparent discrepancy between the archaeological and textual bodies of evidence.
David Ian Lightbody
cf Cooper, L. Did Egyptian Scribes Have an Algorithmic Means for Determining the Circumference of a Circle? http://www.sciencedirect.com/science/article/pii/S0315086011000449
A New Interpretation of Problem 10 of the Moscow Mathematical Papyrus http://www.sciencedirect.com/science/article/pii/S0315086009000305
2011 Did Egyptian Scribes Have an Algorithmic Means for Determining the Circumference of a Circle? Historia Mathematica 38: 455-484.
Edwards, I. E. S.
1979 The Pyramids of Egypt. Middlesex: Penguin. Gillings, R. J. 1982 Mathematics in the Time of the Pharaohs. New York: Dover.
Legon, J. A. R.
1990 The 14 to 11 proportion at Meydum. Discussions in Egyptology 17: 15-22.
Legon, J. A. R.
1991 On Pyramid Dimensions and Proportions. Discussions in Egyptology 20: 35-44.
2008 Egyptian Tomb Architecture. The Archaeological Facts of Pharaonic Circular Symbolism. Oxford: British Archaeological Reports International Series. S1852.
2012 The Encircling Protection of Horus. Proceedings of the Twelfth Annual Symposium Current Researches in Egyptology, 2011, University of Durham: 133-140.
Petrie, W. M. F.
1892 Medum. London: David Nutt, 270, 271, Strand.
Petrie, W. M. F.
1925 Surveys of the Great Pyramids. Nature: 942-943.
Petrie, W. M. F.
1940 Wisdom of the Egyptians. London: British School of Archaeology in Egypt and B. Quaritch Ltd.
2003 The Pyramids: Their Archaeology and History. London: Atlantic Books.